From Hypothetical Strings to Measurable Waves

Cosmic strings are theoretical one-dimensional defects that may have formed during symmetry-breaking phase transitions in the early universe. While no direct evidence for their existence has yet been found, they remain a captivating possibility in cosmology and high-energy physics. Their enormous tension could seed distinctive gravitational phenomena, and if they form closed loops, those loops would oscillate and radiate gravitational waves. This calculator models the power emitted by such a loop and how long the loop would survive before radiating away its energy. Even though the tool is speculative, it provides a playground for exploring the rich interplay of fundamental constants, topological defects, and gravitational radiation.

In the simplest picture, a cosmic string carries a constant mass per unit length μ, often expressed through the dimensionless combination $\frac{G}{c}\mu$. This quantity, written as Gμ/c² and typically denoted simply as Gμ, encapsulates how strongly the string's gravitational effects couple to spacetime. Observational constraints from the cosmic microwave background and pulsar timing suggest Gμ/c² ≤ 10^-7 for many models, hence the default value in the calculator. When a loop forms, its total energy is approximately $E=\mu L{c}^2$, where L is the loop's length. Because cosmic strings are relativistic objects, they can oscillate with velocities approaching the speed of light, shedding energy in the form of gravitational waves with power $P=\Gamma G{\mu }^{2}c$. The dimensionless constant Γ ≈ 50 arises from simulations that estimate how efficiently loops radiate through their various oscillatory modes.

These relationships yield a loop lifetime $\tau =\frac{E}{P}=\frac{L}{\Gamma }$. Remarkably, the power is independent of loop length, while the lifetime scales linearly with L. Because the dimensionless tension $\frac{\mathrm{G\mu }}{c^2}$ has no units, the physical mass per unit length is recovered via $\mu =\frac{\mathrm{G\mu }}{c^2}/G$. Once μ is known in SI units (kg/m), energy, power, and lifetime follow straightforwardly. By experimenting with Gμ values approaching current observational bounds, you can explore how powerful gravitational-wave emission would be and whether future detectors might glimpse such exotic sources.

Sample Results

The table below shows illustrative outputs for different loop lengths assuming Gμ/c² = 10^-7 and Γ = 50. The lifetime is given in years for readability, highlighting how rapidly small loops evaporate compared to macroscopic ones.

Loop Length L (m)	Power P (W)	Lifetime τ (years)
10-3	1.2×1039	5.3×10-29
103	1.2×1039	5.3×10-23
109	1.2×1039	5.3×10-17
1015	1.2×1039	5.3×10-11

Note that the power remains constant because it depends only on the tension parameter, while the lifetime scales with L. Realistic cosmic string loops could range from microscopic to cosmological sizes, and their gravitational-wave signatures would span frequencies from kilohertz to nanohertz, potentially observable by detectors such as LIGO, LISA, or pulsar timing arrays.

How the Calculator Works

When you enter a dimensionless tension, loop length, and emission efficiency, the calculator performs several steps:

1. Convert the dimensionless tension x = Gμ/c² into a mass per unit length μ = x c²/G.
2. Compute total loop energy E = μ L c².
3. Calculate gravitational-wave power P = Γ G μ² c.
4. Determine the lifetime τ = E/P = L/( Γ G μ / c ).
5. Format the results for display, giving power in watts and lifetime in both seconds and years.

The underlying constants are the speed of light c = 299,792,458 m/s and the gravitational constant G = 6.67430×10^-11 m³/kg/s². All arithmetic is done in double-precision JavaScript numbers, which suffices for the orders of magnitude considered here.

Context and Implications

If cosmic strings exist, they could leave numerous imprints on cosmological observables: discontinuities in the cosmic microwave background, gravitational lensing events, bursts of high-frequency gravitational waves, and even seed structure formation. Loops are particularly interesting because their oscillations may produce a stochastic gravitational-wave background. By adjusting the tension parameter, you can explore how sensitive experiments must be to detect such signals. For example, pulsar timing arrays have already ruled out certain Gμ ranges by searching for nanohertz backgrounds. Future detectors might push these limits, potentially revealing faint whispers of early-universe physics.

The lifetime estimate also hints at the population of loops today. Short-lived loops decay quickly after formation, whereas long loops could persist, contributing to the gravitational-wave background. Some scenarios predict a scaling network where loops continually form and decay, maintaining a steady energy density fraction. Understanding these dynamics requires complex numerical simulations, but simple calculators like this one offer intuition about characteristic scales.

Limitations and Speculative Nature

This calculator is necessarily idealized. Real cosmic string loops may develop kinks or cusps that enhance radiation, deviate from perfect relativistic motion, or interact with surrounding matter. The emission efficiency Γ can vary depending on loop shape and gravitational backreaction. Some models include additional radiation channels, such as scalar or gauge bosons, which would shorten the lifetime further. Moreover, the theoretical underpinnings of cosmic strings span grand unified theories, superstring theory, and other speculative frameworks; each introduces parameters and uncertainties far beyond the scope of this tool.

Despite these caveats, the calculator offers a sandbox for imagination. By manipulating extreme tensions or enormous loop lengths, you can conceive scenarios ranging from microscopic vibrations to galaxy-spanning filaments. Though the numbers may be fantastical, they underscore the enormous energies involved and the potential observability of these objects if they populate the universe.

Using the Calculator

Enter your desired tension, loop length, and emission efficiency, then click “Compute Power.” The output displays the mass per unit length, total energy, gravitational-wave power, and loop lifetime. Because results can span many orders of magnitude, scientific notation is used for clarity. All calculations are performed locally within your browser, preserving privacy and enabling experimentation even without internet access. The tool is intended for educational and speculative exploration of cosmic string phenomenology.

By providing an accessible interface to high-energy concepts, the calculator aims to demystify an exotic topic. Whether cosmic strings remain a theoretical curiosity or emerge as a real component of our universe, pondering their consequences broadens our appreciation of physics at the intersection of cosmology, relativity, and topology.